Equation Of A Line Passing Through (-3,-1) And (5,5)
In mathematics, determining the equation of a line is a fundamental concept. Given two points, we can uniquely define a straight line that passes through them. This article will delve into the process of finding the equation of a line when two points are provided, specifically the points (-3, -1) and (5, 5). We will explore the underlying principles, the step-by-step methodology, and provide a clear understanding of the concepts involved. Understanding how to find the equation of a line is crucial in various fields, including algebra, geometry, calculus, and even practical applications in physics, engineering, and computer graphics. This article aims to provide a comprehensive guide that will help you master this essential skill.
Understanding the Basics
Before diving into the specific problem, let's review some essential concepts related to linear equations. The equation of a line can be represented in several forms, but the most common and widely used is the slope-intercept form, which is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope, denoted by m, measures the steepness of the line and is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The y-intercept, denoted by b, is the point where the line intersects the y-axis. Another useful form is the point-slope form, which is given by y - y1 = m(x - x1), where (x1, y1) is a known point on the line and m is the slope. This form is particularly helpful when we have a point and the slope, or when we can calculate the slope from two given points. Understanding these basic forms is crucial for finding the equation of a line in different scenarios. Each form provides a unique way to represent the line, and choosing the appropriate form can simplify the process of finding the equation. The slope-intercept form is excellent for visualizing the line's behavior, while the point-slope form is more direct when given a point and slope. Familiarity with both forms provides a versatile toolkit for solving linear equation problems.
Calculating the Slope
The slope is a crucial parameter that defines the direction and steepness of a line. To find the slope (m) of a line passing through two points, say (-3, -1) and (5, 5), we use the formula:
m = (y2 - y1) / (x2 - x1)
Here, (-3, -1) can be considered as (x1, y1) and (5, 5) as (x2, y2). Substituting these values into the formula, we get:
m = (5 - (-1)) / (5 - (-3))
Simplifying the expression, we have:
m = (5 + 1) / (5 + 3)
m = 6 / 8
Reducing the fraction, we find:
m = 3 / 4
Therefore, the slope of the line passing through the points (-3, -1) and (5, 5) is 3/4. This means that for every 4 units we move horizontally along the line, we move 3 units vertically. The positive slope indicates that the line is increasing, meaning it rises as we move from left to right. This calculation of the slope is the foundation for determining the equation of the line, as it provides the rate of change of y with respect to x. Understanding this concept is crucial for further steps in finding the line's equation. The slope provides a numerical measure of the line's inclination and is a fundamental property used in various mathematical and real-world applications. Accurate calculation of the slope is essential for correctly representing the line's behavior.
Using the Point-Slope Form
Now that we have calculated the slope (m = 3/4), we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope. We can use either of the given points, (-3, -1) or (5, 5), as (x1, y1). Let's use the point (-3, -1). Substituting the values into the point-slope form, we get:
y - (-1) = (3/4)(x - (-3))
Simplifying the equation:
y + 1 = (3/4)(x + 3)
This is the equation of the line in point-slope form. This form directly represents the line using a known point and the slope. While it's a valid representation, it's often useful to convert it into slope-intercept form for easier interpretation and comparison with other lines. The point-slope form is particularly advantageous when you have a point and the slope, as it allows you to directly construct the equation without needing to solve for the y-intercept initially. It highlights the relationship between the coordinates of any point on the line and the slope, providing a clear geometric interpretation. The flexibility of choosing either of the given points allows for a verification of the final result, as both points should lead to the same equation in slope-intercept form after simplification.
Converting to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form (y = mx + b), we need to isolate y. Starting from the equation we derived in the previous step:
y + 1 = (3/4)(x + 3)
First, distribute the (3/4) on the right side of the equation:
y + 1 = (3/4)x + (3/4)(3)
y + 1 = (3/4)x + 9/4
Next, subtract 1 from both sides of the equation to isolate y:
y = (3/4)x + 9/4 - 1
To combine the constants, we need a common denominator. Since 1 can be written as 4/4, the equation becomes:
y = (3/4)x + 9/4 - 4/4
y = (3/4)x + 5/4
This is the equation of the line in slope-intercept form. We can see that the slope m is 3/4 and the y-intercept b is 5/4. This form provides a clear visualization of the line's behavior, showing its slope and where it intersects the y-axis. Converting to slope-intercept form is a crucial step in understanding and analyzing linear equations, as it readily provides key information about the line. The process of distribution and simplification ensures that the equation is in its simplest and most easily interpretable form. This form is also useful for graphing the line and comparing it with other lines. The ability to switch between point-slope and slope-intercept forms is a fundamental skill in linear algebra.
Verification
To ensure the equation we found is correct, we can verify that both given points, (-3, -1) and (5, 5), satisfy the equation y = (3/4)x + 5/4.
Let's substitute the first point (-3, -1) into the equation:
-1 = (3/4)(-3) + 5/4
-1 = -9/4 + 5/4
-1 = -4/4
-1 = -1
The equation holds true for the first point. Now, let's substitute the second point (5, 5) into the equation:
5 = (3/4)(5) + 5/4
5 = 15/4 + 5/4
5 = 20/4
5 = 5
The equation also holds true for the second point. Since both points satisfy the equation y = (3/4)x + 5/4, we can confidently say that this is the correct equation of the line passing through the points (-3, -1) and (5, 5). This verification process is a crucial step in problem-solving, as it confirms the accuracy of the solution. By substituting the given points back into the equation, we ensure that the derived equation correctly represents the line. This method provides a simple and effective way to check the solution's validity. It reinforces the understanding of the relationship between the points on a line and its equation. The consistency of the results for both points strengthens the confidence in the final answer.
Conclusion
In this article, we have successfully determined the equation of the line that passes through the points (-3, -1) and (5, 5). We began by understanding the basic forms of linear equations, including the slope-intercept form and the point-slope form. We then calculated the slope of the line using the formula m = (y2 - y1) / (x2 - x1). After finding the slope, we used the point-slope form to derive an initial equation of the line. We further converted this equation into slope-intercept form, y = (3/4)x + 5/4, which provides a clear representation of the line's slope and y-intercept. Finally, we verified our solution by substituting the original points into the equation and confirming that they satisfy it. This process demonstrates a systematic approach to finding the equation of a line given two points. The steps involved highlight the importance of understanding the underlying concepts and applying the appropriate formulas. Mastering this skill is essential for various mathematical applications and problem-solving scenarios. The ability to confidently find the equation of a line is a fundamental building block in algebra and geometry, paving the way for more advanced topics and applications in diverse fields. The clear and step-by-step methodology presented in this article provides a solid foundation for further exploration and practice in linear equations.
